具有非零边界条件的非齐次五阶非线性薛定谔方程的Riemann-Hilbert逼近
Riemann-Hilbert approach to the inhomogeneous fifth-order nonlinear Schrödinger equation with non-vanishing boundary conditions
摘要：
详细研究了具有非零边界条件的非齐次五阶非线性薛定谔方程。首先，对散射问题进行了谱分析。首先引入黎曼曲面和仿射参数，将原谱参数转换为新参数，避免了多值问题。基于ifoNLS方程的拉克斯对，得到了Jost函数，系统地建立了它们的解析性质、渐近性质、对称性质以及散射矩阵的相应性质.对于逆散射问题，我们分别讨论了散射系数具有单零点和双零点的情形，并进一步导出了相应的精确解。此外，我们还发现了一些有趣的现象，当我们为这些精确解选择适当的参数时，这有助于研究这些解的传播行为。
英文摘要：
We consider the inhomogeneous fifth-order nonlinear Schrödinger (ifoNLS) equation with nonzero boundary condition in detailed. Firstly, the spectral analysis of the scattering problem is carried out. A Riemann surface and affine parameters are first introduced to transform the original spectral parameter to a new parameter in order to avoid the multi-valued problem. Based on Lax pair of the ifoNLS equation, the Jost functions are obtained, and their analytical, asymptotic, symmetric properties, as well as the corresponding properties of the scattering matrix are established systematically. For the inverse scattering problem, we discuss the cases that the scattering coefficients have simple zeros and double zeros, respectively, and we further derive their corresponding exact solutions. Moreover, some interesting phenomena are found when we choose some appropriate parameters for these exact solutions, which is helpful to study the propagation behavior of these solutions.
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